Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 491357, 13 pages
doi:10.1155/2010/491357
Research Article
Strong Convergence Theorems for
Equilibrium Problems and Fixed Point Problems in
Hilbert Spaces
Jian-Wen Peng1 and Yan Wang2
1
2
College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China
Division of Mathematics, Yibin No. 3 Middle School, Yibin, Sichuan 644000, China
Correspondence should be addressed to Jian-Wen Peng, jwpeng6@yahoo.com.cn
Received 21 August 2009; Revised 1 December 2009; Accepted 13 January 2010
Academic Editor: Petru Jebelean
Copyright q 2010 J.-W. Peng and Y. Wang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We introduce an Ishikawa iterative scheme by the viscosity approximate method for finding
a common element of the set of solutions of an equilibrium problem and the set of fixed points
of a nonexpansive mapping in Hilbert space. Then, we prove some strong convergence theorems
which extend and generalize S. Takahashi and W. Takahashi’s results 2007.
1. Introduction
Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let F be a
bifunction from C × C to R, where R is the set of real numbers. The equilibrium problem for
F : C × C → R is to find x ∈ C such that
F x, y ≥ 0,
∀y ∈ C.
1.1
The set of solutions of 1.1 is denoted by EPF. Given a mapping T : C → H, let
Fx, y T x, y − x for all x, y ∈ C. Then, z ∈ EPF if and only if T z, y − z ≥ 0 for all
y ∈ C. Numerous problems in physics, optimization, and economics reduce to find a solution
of 1.1; for more details, see 1, 2.
Recall that a self-mapping S of a closed convex subset C of H is nonexpansive 3 if
there holds that
Sx − Sy ≤ x − y,
∀x, y ∈ C.
1.2
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We denote the set of fixed points of S by FS. There are some methods for approximation
of fixed points of a nonexpansive mapping. In 2000, Moudafi 4 introduced the viscosity
approximation method for nonexpansive mappings see 5 for further developments in both
Hilbert and Banach spaces. Some methods have been proposed to solve the equilibrium
problem; see, for instance, 1, 2, 6, 7. Recently, Combettes and Hirstoaga 6 introduced an
iterative scheme of finding the best approximation to the initial data when EPF is nonempty
and proved a strong convergence theorem. S. Takahashi and W. Takahashi 7 introduced a
Mann iterative scheme by the viscosity approximation method for finding a common element
of the set of solution 1.1 and the set of fixed points of a nonexpansive mapping in a Hilbert
space and proved a strong convergence theorem.
On the other hand, Ishikawa 8 introduced the following iterative process defined
recursively by
xn1 αn xn 1 − αn Syn ,
yn βn xn 1 − βn Sxn , ∀n ∈ N,
1.3
where the initial guess x0 is taking in C arbitrarily, {αn } and {βn } are sequences in the interval
0, 1.
In this paper, motivated by the ideas in 4–8, we introduce an Ishikawa iterative
scheme by the viscosity approximation method for finding a common element of the set of
solution 1.1 and the set of fixed points of a nonexpansive mapping in a Hilbert space.
Starting with an arbitrary x1 ∈ H, define sequences {xn }, {yn }, and {un } by
1
F un , y y − un , un − xn ≥ 0,
rn
∀y ∈ C,
xn1 αn fxn 1 − αn Syn ,
yn βn xn 1 − βn Sun , ∀n ∈ N,
1.4
where {αn }, {βn } ⊂ 0, 1 and {rn } ⊂ 0, ∞.
We will prove in Section 3 that if the sequences {αn }, {βn }, and {rn } of parameters
satisfy appropriate conditions, then the sequences {xn }, {yn }, and {un } generated by 1.4
converge strongly to z ∈ FS ∩ EPF. The results in this paper extend and generalize S.
Takahashi and W. Takahashi’s results 7.
2. Preliminaries
Let H be a real Hilbert space with inner product ·, ·, and norm · and let C be a nonempty
closed convex subset of H. xn → x implies that {xn } converges strongly to x and xn x
means that {xn } converges weakly to x. In a real Hilbert space H, we have
λx 1 − λy2 λx2 1 − λy2 − λ1 − λx − y2
for all x, y ∈ H and λ ∈ R; see 9.
2.1
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3
For any x ∈ H, there exists a unique nearest point in C, denoted by PC x, such that
x − PC x ≤ x − y for all y ∈ C. Such a PC is called the metric projection of H onto C. It is
also known that y PC x is equivalent to x − y, y − z ≥ 0 for all z ∈ C.
For solving the equilibrium problem, let us assume that the bifunction F satisfies the
following conditions:
A1 Fx, x 0 for all x ∈ C;
A2 F is monotone, that is, Fx, y Fy, x ≤ 0 for any x, y ∈ C;
A3 for each x, y, z ∈ C,
limF tz 1 − tx, y ≤ F x, y ;
2.2
t↓0
A4 for each x ∈ C, y → Fx, y is convex and lower semicontinuous.
We recall some lemmas needed later.
Lemma 2.1 see 2. Let C be a nonempty closed convex subset of H and let F be a bifunction from
C × C to R satisfying (A1)–(A4). Let r > 0 and x ∈ H. Then, there exists z ∈ C such that
1
F z, y y − z, z − x ≥ 0,
r
∀y ∈ C.
2.3
Lemma 2.2 see 5. Let C be a nonempty closed convex subset of H, and let F be a bifunction from
C × C to R satisfying (A1)–(A4). For r > 0 and x ∈ H, define a mapping Tr : H → C as follows:
Tr x 1
z ∈ C : F z, y y − z, z − x ≥ 0, ∀y ∈ C
r
2.4
for all x ∈ H. Then, the following statements hold:
1 Tr is single-valued;
2 Tr is firmly nonexpansive, that is, for any x, y ∈ H,
Tr x − Tr y2 ≤ Tr x − Tr y , x − y ;
2.5
3 FTr EPF;
4 EPF is closed and convex.
Lemma 2.3 see 10. Let {an } be a sequence of nonnegative real numbers such that
an1 ≤ 1 − cn an bn ,
∀n ∈ N,
2.6
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where {bn } is a sequence of real numbers and {cn } is a sequence in 0, 1 such that
i
∞
n1 cn
∞,
ii lim supn → ∞ bn /cn ≤ 0 or
∞
n1
|bn | < ∞.
Then, limn → ∞ an 0.
3. Strong Convergence Theorem
In this section, we show a strong convergence theorem which solves the problem of finding a
common element of the set of solutions of an equilibrium problem and the set of fixed points
of a nonexpansive mapping in a Hilbert space.
Theorem 3.1. Let C be a nonempty closed convex subset of H. Let F be a bifunction from C × C to R
satisfying (A1)–(A4) and let S be a nonexpansive mapping of C into H such that FS ∩ EPF /
∅.
Let f be a contraction of H into itself and let {xn }, {un }, and {yn } be sequences generated by x1 ∈ H
and 1.4. If {αn }, {βn } ⊂ 0, 1 and {rn } ⊂ 0, ∞ satisfy the following conditions:
lim αn 0,
n→∞
∞
αn ∞,
n1
n1
0 < lim inf βn ≤ lim sup βn < 1,
n→∞
n→∞
lim inf rn > 0,
n→∞
∞
|αn1 − αn | < ∞,
∞ βn1 − βn < ∞,
n1
3.1
∞
|rn1 − rn | < ∞,
n1
then, {xn }, {yn }, and {un } converge strongly to z ∈ FS ∩ EPF, where z PFS∩EPF fz.
Proof. Let Q PFS∩EPF . Then Qf is a contraction of H into itself. In fact, there exists a ∈
0, 1 such that fx − fy ≤ ax − y for all x, y ∈ H. So, we have that
Qfx − Qf y ≤ fx − f y ≤ ax − y
3.2
for all x, y ∈ H. Since H is complete, there exists a unique element z ∈ H such that z Qfz.
Such a z ∈ H is an element of C.
Let v ∈ FS ∩ EPF. Then from un Trn xn , we have
un − v Trn xn − Trn v ≤ xn − v
3.3
for all n ∈ N. Put M max{x1 − v, 1/1 − afv − v}. It is obvious that x1 − v ≤ M.
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5
Suppose xn − v ≤ M. Then, we have
xn1 − v ≤ αn fxn − v 1 − αn Syn − v
≤ αn fxn − fv αn fv − v 1 − αn Syn − v
3.4
≤ aαn xn − v αn fv − v 1 − αn yn − v.
On the other hand
yn − v ≤ βn xn − v 1 − βn Sun − v
≤ βn xn − v 1 − βn un − v
3.5
≤ βn xn − v 1 − βn xn − v
xn − v.
Putting 3.5 into 3.4, we have
xn1 − v ≤ aαn xn − v αn fv − v 1 − αn xn − v
fv − v
1 − αn 1 − axn − v αn 1 − a
1−a
3.6
≤ 1 − αn 1 − aM αn 1 − aM M.
So, we have that xn1 − v ≤ M for any n ∈ N. And hence {xn } is bounded. We also obtain
that {un }, {yn }, {Sun }, {Syn }, and {fxn } are bounded. Next, we show that limn → ∞ xn1 −
xn 0. In fact,
yn − yn−1 βn xn 1 − βn Sun − βn−1 xn−1 1 − βn−1 Sun−1 βn xn − xn−1 βn − βn−1 xn−1 1 − βn Sun − Sun−1 βn−1 − βn Sun−1 ≤ βn − βn−1 xn−1 βn xn − xn−1 1 − βn un − un−1 βn − βn−1 Sun−1 ,
3.7
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International Journal of Mathematics and Mathematical Sciences
and hence
xn1 − xn αn fxn 1 − αn Syn − αn−1 fxn−1 − 1 − αn−1 Syn−1 αn fxn − αn fxn−1 αn fxn−1 − αn−1 fxn−1 1 − αn Syn − 1 − αn Syn−1 1 − αn Syn−1 − 1 − αn−1 Syn−1 ≤ αn axn − xn−1 |αn − αn−1 |fxn−1 1 − αn yn − yn−1 |αn − αn−1 |Syn−1 ≤ αn axn − xn−1 |αn − αn−1 |fxn−1 1 − αn × βn − βn−1 xn−1 βn xn − xn−1 1 − βn un − un−1 βn − βn−1 Sun−1 |αn − αn−1 |Syn−1 βn − αn βn − a xn − xn−1 |αn − αn−1 | fxn−1 Syn−1 1 − αn βn − βn−1 xn−1 Sun−1 1 − αn 1 − βn un − un−1 ≤ βn − αn βn − a xn − xn−1 |αn − αn−1 |K1 1 − αn βn − βn−1 K2
1 − αn 1 − βn un − un−1 ,
3.8
where K1 sup{fxn Syn : n ∈ N} and K2 sup{xn Sun : n ∈ N}.
On the other hand, from un Trn xn and un1 Trn1 xn1 , we have
1
F un , y y − un , un − xn ≥ 0,
rn
∀y ∈ C,
1 F un1 , y y − un1 , un1 − xn1 ≥ 0,
rn1
∀y ∈ C.
3.9
3.10
Putting y un1 in 3.9 and y un in 3.10, we have
Fun , un1 Fun1 , un 1
un1 − un , un − xn ≥ 0,
rn
1
rn1
3.11
un − un1 , un1 − xn1 ≥ 0.
So, from the monotonicity of F, we get
un − xn un1 − xn1
un1 − un ,
−
≥ 0,
rn
rn1
3.12
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7
and hence
rn
un1 − un , un − un1 un1 − xn −
un1 − xn1 ≥ 0.
rn1
3.13
Without loss of generality, let us assume that there exists a real number b such that rn > b > 0
for all n ∈ N. Then, we have
rn
un1 − un , xn1 − xn 1 −
un1 − xn1 rn1
rn −
x
≤ un1 − un xn1 − xn 1 −
,
u
n1
n1
rn1 un1 − un 2 ≤
3.14
and hence
un1 − un ≤ xn1 − xn 1
|rn1 − rn |un1 − xn1 rn1
1
≤ xn1 − xn |rn1 − rn |L,
b
3.15
where L sup{un − xn : n ∈ N}. So from 3.8, we have
xn1 − xn ≤ βn − αn βn − a xn − xn−1 |αn − αn−1 |K1
1
1 − αn βn − βn−1 K2 1 − αn 1 − βn xn − xn−1 |rn − rn−1 |L
b
1 − αn 1 − axn − xn−1 |αn − αn−1 |K1
1
1 − αn βn − βn−1 K2 1 − αn 1 − βn |rn − rn−1 |L.
b
3.16
Using Lemma 2.1 in 10, we obtain
lim xn1 − xn 0.
n→∞
3.17
From 3.15 and |rn1 − rn | → 0, we have
n→∞
lim un1 − un 0.
3.18
lim yn1 − yn 0.
3.19
It follows from 3.7 that
n→∞
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Since xn αn−1 fxn−1 1 − αn−1 Syn−1 , we have
xn − Syn ≤ xn − Syn−1 Syn−1 − Syn ≤ αn−1 fxn−1 − Syn−1 yn−1 − yn .
3.20
From αn → 0, we have xn − Syn → 0. For v ∈ FS ∩ EPF, we have
un − v2 Trn xn − Trn v2
≤ Trn xn − Trn v, xn − v
un − v, xn − v
3.21
1
un − v2 xn − v2 − xn − un 2 ,
2
and hence
un − v2 ≤ xn − v2 − xn − un 2 .
3.22
Therefore, from the convexity of · 2 , we have
yn − v2 ≤ βn xn − v2 1 − βn Sun − v2
≤ βn xn − v2 1 − βn un − v2
≤ βn xn − v2 1 − βn xn − v2 − xn − un 2
3.23
xn − v2 − 1 − βn xn − un 2 ,
and hence
2
xn1 − v2 αn fxn 1 − αn Syn − v
2
2
≤ αn fxn − v 1 − αn Syn − v
2
2
≤ αn fxn − v 1 − αn yn − v
2
≤ αn fxn − v 1 − αn xn − v2 − 1 − βn xn − un 2
3.24
2
≤ αn fxn − v xn − v2 − 1 − αn 1 − βn xn − un 2 .
So, we have
2
1 − αn 1 − βn xn − un 2 ≤ αn fxn − v xn − v2 − xn1 − v2
2
≤ αn fxn − v xn − xn1 xn − v xn1 − v.
3.25
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Without loss of generality, let us assume that there exists two real numbers β∗ and β such that
1 > β ≥ βn ≥ β∗ > 0 for all n ∈ N. Hence,
1 − αn 1 − β xn − un 2 ≤ 1 − αn 1 − βn xn − un 2
2
≤ αn fxn − v xn − xn1 xn − v xn1 − v.
3.26
It follows that xn − un → 0. We also have
Sun − xn ≤ Syn − xn Sun − Syn ≤ Syn − xn un − yn ≤ Syn − xn un − xn xn − yn Syn − xn un − xn 1 − βn xn − Sun .
3.27
β∗ Sun − xn ≤ βn Sun − xn ≤ Syn − xn un − xn .
3.28
It follows that
Hence, Sun − xn → 0. Since
Sun − un ≤ Sun − xn xn − un ,
3.29
we also have limn → ∞ Sun − un 0. Next, we show that
lim sup fz − z, xn − z ≤ 0,
3.30
n→∞
where z PFS∩EPF fz. To show this inequality, we choose a subsequence {xni } of {xn }
such that
lim fz − z, xni − z lim sup fz − z, xn − z .
n→∞
n→∞
3.31
Since {uni } is bounded, there exists a subsequence {uni j } of {uni } which converges weakly
to w. Without loss of generality, we can assume that {uni } w. From Sun − un → 0, we
obtain Suni w. Let us show w ∈ EPF. By un Trn xn , we have
1
F un , y y − un , un − xn ≥ 0,
rn
∀y ∈ C.
3.32
From A2, we also have
1
y − un , un − xn ≥ F y, un ,
rn
3.33
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and hence,
y − uni ,
uni − xni
rni
≥ F y, uni .
3.34
Since uni − xni /rni → 0 and uni w, from A4, we have
f y, w ≤ 0,
∀y ∈ C.
3.35
For t with 0 < t ≤ 1 and y ∈ C, let yt ty 1 − tw. Since y ∈ C and w ∈ C, we obtain yt ∈ C
and hence Fyt , w ≤ 0. So we have
0 F yt , yt ≤ tF yt , y 1 − tF yt , w ≤ tF yt , y .
3.36
Dividing by t, we get
F yt , y ≥ 0.
3.37
F w, y ≥ 0
3.38
Letting t → 0 and from A3, we get
for all y ∈ C and hence w ∈ EPF. We shall show that w ∈ FS. Assume w /
∈ FS. Since
uni w and w /
Sw, from the Opial theorem 11 we have
lim infuni − w < lim infuni − Sw
i→∞
i→∞
≤ lim inf{uni − Suni Suni − Sw}
i→∞
3.39
≤ lim infuni − w.
i→∞
This is a contradiction. So, we get w ∈ FS. Therefore, w ∈ FS ∩ EPF. Since z PFS∩EPF fz, we have
lim sup fz − z, xn − z lim fz − z, xni − z
n→∞
i→∞
lim fz − z, uni − z
i→∞
fz − z, w − z ≤ 0.
3.40
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11
From xn1 − z αn fxn − z 1 − αn Syn − z, we have
2
1 − αn 2 Syn − z ≥ xn1 − z2 − 2αn fxn − z, xn1 − z ,
yn − z2 βn xn 1 − βn Sun − z2
≤ βn xn − z2 1 − βn Sun − z2
≤ βn xn − z2 1 − βn un − z2
3.41
3.42
≤ xn − z2 .
It follows that
2
xn1 − z2 ≤ 1 − αn 2 Syn − z 2αn fxn − z, xn1 − z
2
≤ 1 − αn 2 yn − z 2αn fxn − fz, xn1 − z
2αn fz − z, xn1 − z
≤ 1 − αn 2 xn − z2 2αn axn − zxn1 − z
3.43
2αn fz − z, xn1 − z
≤ 1 − αn 2 xn − z2 αn a xn − z2 xn1 − z2
2αn fz − z, xn1 − z .
Hence
xn1 − z2 ≤
2αn 1 − αn 2 αn a
fz − z, xn1 − z .
xn − z2 1 − αn a
1 − αn a
3.44
From αn → 0, we know that there exists a positive integer n0 , such that 1 > 1 − αn a > 1/2 for
all n ≥ n0 . Then
α2n
1 − αn 2 αn a 1 − 2αn αn a
1 − αn a
1 − αn a
1 − αn a
1−
α2n
21 − aαn
1 − αn a
1 − αn a
≤ 1 − 21 − aαn 2α2n ,
∀n ≥ n0 .
3.45
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International Journal of Mathematics and Mathematical Sciences
Putting above inequality into 3.44, we get
xn1 − z2 ≤ 1 − 21 − aαn xn − z2 2Mα2n 2αn
σn ,
1 − αn a
∀n ≥ n0 ,
3.46
where M sup{xn − z2 : n ∈ N}, and σn fz − z, xn1 − z.
It follows from Lemma 2.3 that
xn −→ z ∈ FS ∩ EPF.
3.47
It follows from xn − un → 0 and 3.42 that un → z and yn → z.
By Theorem 3.1, we can obtain the following new result.
Corollary 3.2. Let C be a nonempty closed convex subset of H. Let S be a nonexpansive mapping of C
into H such that FS /
∅. Let f be a contraction of H into itself and let {xn } and {yn } be sequences
generated by x1 ∈ H and
xn1 αn fxn 1 − αn Syn ,
yn βn xn 1 − βn SPC xn , ∀n ∈ N.
3.48
If {αn }, {βn } ⊂ 0, 1 satisfy the following conditions:
lim αn 0,
n→∞
∞
αn ∞,
n1
0 < lim infβn ≤ lim supβn < 1,
n→∞
n→∞
∞
|αn1 − αn | < ∞,
n1
∞ βn1 − βn < ∞,
3.49
n1
then, {xn } and {yn } converge strongly to z ∈ FS, where z PFS fz.
Proof. Put Fx, y 0 for all x, y ∈ C and rn 1 for all n ∈ N in Theorem 3.1. Then, we get
un PC xn . So from Theorem 3.1, the sequences {xn } and {yn } converge strongly to z ∈ FS,
where z PFS fz.
Remark 3.3. Theorem 3.1 and Corollary 3.2, respectively, extend and generalize Theorem 3.2
and Corollary 3.3 in 7 from the Mann iterative form to the Ishikawa iterative form.
Acknowledgments
The authors would like to express their thanks to the referees for helpful suggestions.
This research was supported by the National Natural Science Foundation of China Grants
10771228 and 10831009, the Natural Science Foundation of Chongqing Grant no. CSTC,
2009BB8240, and the Research Project of Chongqing Normal University Grant 08XLZ05.
International Journal of Mathematics and Mathematical Sciences
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